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Unformatted text preview: LS. LINEAR SYSTEMS LS.1 Review of Linear Algebra In these notes, we will investigate a way of handling a linear system of ODEs directly, instead of using elimination to reduce it to a single higher-order equation. This gives im- portant new insights into such systems, and it is usually a more convenient and faster way of solving them. The method makes use of some elementary ideas about linear algebra and matrices, which we will assume you know from your work in multivariable calculus. Your textbook contains a section (5.3) reviewing most of these facts, with numerical examples. Another source is the 18.02 Supplementary Notes, which contains a beginning section on linear algebra covering approximately the right material. For your convenience, what you need to know is summarized briey in this section. Consult the above references for more details and for numerical examples. 1. Vectors. A vector (or n-vector ) is an n-tuple of numbers; they are usually real numbers, but we will sometimes allow them to be complex numbers, and all the rules and operations below apply just as well to n-tuples of complex numbers. (In the context of vectors, a single real or complex number, i.e., a constant, is called a scalar .) The n-tuple can be written horizontally as a row vector or vertically as a column vector . In these notes it will almost always be a column. To save space, we will sometimes write the column vector as shown below; the small T stands for transpose , and means: change the row to a column. a = ( a 1 , . . . , a n ) row vector a = ( a 1 , . . . , a n ) T column vector These notes use boldface for vectors; in handwriting, place an arrow a over the letter. Vector operations. The three standard operations on n-vectors are: addition: ( a 1 , . . . , a n ) + ( b 1 , . . . , b n ) = ( a 1 + b 1 , . . . , a n + b n ) . multiplication by a scalar : c ( a 1 , . . . , a n ) = ( ca 1 , . . . , ca n ) scalar product : ( a 1 , . . . , a n ) ( b 1 , . . . , b n ) = a 1 b 1 + . . . + a n b n . 2. Matrices and Determinants. An m n matrix A is a rectangular array of numbers (real or complex) having m rows and n columns. The element in the i-th row and j-th column is called the ij-th entry and written a ij . The matrix itself is sometimes written ( a ij ), i.e., by giving its generic entry, inside the matrix parentheses. A 1 n matrix is a row vector; an n 1 matrix is a column vector....
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